ABSTRACT
Infinite-dimensional Hilbert spaces are of utmost importance for scattering theory and this leads to the following critical question: is the central operator i.e. the Hamiltonian Hˆ, bounded ? As seen in chapter 3, boundedness is the most essential feature of operators defined in infinite-dimensional vector spaces. The answer to this key question is in the negative. The Hamilton operator belongs to a class of so-called positive operators that are also called semi-bounded operators. Stated more precisely, due to the existence of the minimum of the total energy of the investigated system, the Hamiltonian Hˆ is an operator which is bounded from below. This is mathematically expressed by the following requirement for non-negativity of the expected values of the Hamiltonian operator
E = 〈Ψ|HˆΨ〉 ≥ 0 ∀Ψ ∈ DHˆ (5.1)
where DHˆ ⊂ H is the domain i.e. the definition region of the operator Hˆ which represents a closed linear manifold in H. Here, we emphasize the especially important fact that DHˆ is not equal to the whole space H. Rather, DHˆ coincides only with a subspace of H. However, this latter subspace is everywhere dense in H. The notion of a linear manifold is in close connection with the customary notion of a subspace of a given vector space. Let us denote by ψ the elements of the vector space H and by ψ′ the elements of the corresponding subspace H′ ⊂ H. Then the set of all the vectors ψ + ψ′ (ψ ∈ H, ψ′ ∈ H′) is called a linear manifold and it is denoted as ψ +H′
H′ + ψ′′ = {ψ ∈ H : ψ = ψ′ + ψ′′ , ψ′ ∈ H′ , ∃ψ′′ ∈ H} . (5.2)
From the obvious geometric associations, a linear manifold ψ + H′ is also called the translation of the subspace H′ for the vector ψ. For each element ψ ∈ H, the linear manifold ψ + H′ is determined entirely. Moreover, the element ψ ∈ H simultaneously belongs to the linear manifold ψ+H′. Namely, since H′ is a subspace, it follows that ∅ ∈ H, so that ψ = (ψ+∅) ∈ ψ+H′. The relation (5.1) expresses the most important feature of all physical systems and that is the fact the total energy E possesses a lower limit which corresponds
ION-ATOM
to the lowest (E ≡ Emin) i.e. bound state of the system. Nevertheless, the Hamilton operator Hˆ is not bounded from above. This implies that, from the mathematical viewpoint, the Hamiltonian Hˆ is an unbounded operator. Notice that from the physical viewpoint, it might appear at first glance that it could be more acceptable to write E = 〈Ψ|HˆΨ〉 ≥ Emin instead of (5.1). This is because in the inequality (5.1) we arbitrarily assigned the zero value as the lower limit of the energy E. But this does not represent any essential restriction, since the appearance of any additive constant in the Hamiltonian Hˆ would have no physical implication. Namely, an extra constant term in Hˆ could only lead to a scaling on the energy axis where the starting value of the physical energy is chosen arbitrarily anyway. In other words, the expression (5.1) is justified, since the lower limit as the zero limit for the energy can always be obtained by a convenient choice of a constant term in Hˆ0 and Hˆ.
